1. Randomized Algorithms
CS648
Lecture 4
• Linearity of Expectation with applications
(Most important tool for analyzing randomized algorithms)
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2. RECAP FROM THE LAST LECTURE
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3. Random variable
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4. Expected Value of a random variable
(average value)
X= c
Ω
X= a
X= b
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5. Examples
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6. Can we solve these problems ?
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7. Balls into Bins
(number of empty bins)
1 2 3 4 5
1
2
3
…
…
m-1 m
…
This is a right but useless answer !
n
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8. Randomized Quick Sort
(number of comparisons)
A recursion tree
associated with Randomized Quick Sort
We can not proceed from this point …
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9. Balls into Bins
(number of empty bins)
1 2 3 4 5
1
2
3
…
…
m-1 m
…
n
Randomized Quick Sort
(number of comparisons)
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10. Balls into Bins
(number of empty bins)
1 2 3 4 5
…
m-1 m
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11. 1 2 3 4 5 6
1
0
2
1
3
4
0
5
1
0
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12. Sum of Random Variables
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13. Randomized Quick Sort
(number of comparisons)
Elements of A arranged in
Increasing order of values
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14. 1
0
…
0
1
1
…
0
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15. What have we learnt till now?
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16. The main question ?
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17. Balls into Bins
(number of empty bins)
1 2 3 4 5
1
2
3
…
…
m-1 m
…
n
17

18. Randomized Quick Sort
(number of comparisons)
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19. Linearity of Expectation
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20. Where to use Linearity of expectation ?
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21. Think over the following questions?
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22. Think over the following questions?
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23. Independent random variables
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24. Some Practice problems
as homework
• Balls into bin problem:
• What is the expected number of bins having exactly 2 balls ?
• We toss a coin n times, what is the expected number of times pattern HHT
appear ?
• A stick has n joints. The stick is dropped on floor and in this process each
joint may break with probability p independent of others. As a result the
stick will be break into many substicks.
– What is the expected number of substicks of length 3 ?
– What is the expected number of all the substicks ?
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25. PROBLEMS OF THE NEXT LECTURE
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26. Fingerprinting Techniques
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27. Fingerprinting Techniques
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